Tutorial

Transformations of Functions

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Introduction

Transformations of functions allow one to obtain the graph of a function quickly and efficiently if it can be identified as being related to a basic function whose graph is well known. This is because there is a basic visual resemblance between the graph of a basic function and those of its transformations. Although there are many types of transformations, we shall concern ourselves with three basic types: shifts, also known as translations, which involve moving the graph of a basic function vertically, horizontally, or both; reflections, which involve creating a mirror image of a function's graph about the x- or y-axes, or both; and deformations, which involve the horizontal or vertical stretching or shrinking of a graph.

Shifts and (in most cases) reflections both change the position of a graph, but not its shape. Reflections also change the graph's orientation (except in certain cases when the original graph has a symmetry property). Deformations do not change a graph's position or orientation; rather, they change the graph's aspect ratio and thus its shape, causing it to appear shorter, taller, narrower, or wider than the graph of the basic function to which it is related.

More complex transformations may be achieved by performing two or more of the above transformations in sequence. Therefore, they are often referred to as "multiple transformations" or "sequential transformations".

A small set of relatively simple rules governs how certain modifications to the definitions of basic functions are expressed visually when the modified functions are graphed. These rules are clarified in the following discussion, which more closely examines each type of transformation described above. The rules work in a consistent manner when applied to the definition of any function whose domain and range are subsets (including improper subsets) of the real numbers.

Shifts or Translations

The effect of a simple shift or translation is to cause the graph of the translated function to move vertically or horizontally relative to the graph of the basic function to which it is related. For example, the graph of the basic function $$f(x)=x^2$$
is an upward-opening parabola whose vertex lies at the origin (the point (0, 0)) of a Cartesian coordinate system. To move this graph vertically upward by 3 units (so that the vertex lies at (0, 3)) without rotating or deforming it, we change the function to $$g(x)=x^2+3=f(x)+3.$$
To move this graph horizontally rightward by 4 units (so that the vertex lies at (4, 0)), we change the function to $$h(x)={(x-4)}^2=f(x-4)\mathrm{.}$$

In the following instructions, we assume that $f(x)$ represents the original function, and $g(x)$ represents the shifted or translated function.

Cookbook for vertical shifts

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To translate the graph of this function vertically upward by $c$ units, replace "$y$" with "$y-c$".
3. To translate a function vertically downward, use a negative number for $c$, then follow the procedure for an upward shift. (For example, to move 2 units downward, use $c=-2$.)
4. After making the substitution, solve the resulting expression for $y$ (that is, isolate the term $y$ on one side of the resulting equation).
5. Finally, replace "$y$" with "$g(x)$".

Example: Move the graph of the square root function $f(x)=\sqrt{x}$ vertically downward by 4 units.

1. Substitute "$y$" for "$f(x)$": $$y=\sqrt{x}\mathrm{.}$$
   
2. Replace "$y$" with "$y-c$": $$y-c=\sqrt{x}\mathrm{.}$$
   
3. Since the shift is downward, set $c=-4$ and substitute: $$y-(-4)=\sqrt{x}\mathrm{.}$$
   
4. Simplify and solve for $y$: $$y+4=\sqrt{x}\to y=\sqrt{x}-4.$$
   
5. Replace "$y$" with "$g(x)$": $$g(x)=\sqrt{x}-4.$$
   This is the equation of the shifted function. Since $f(x)=\sqrt{x}$, we may also write this in the form $$g(x)=f(x)-\mathrm{4,}$$
   which shows how a generic downward shift may be written in terms of the original and shifted functions $f(x)$ and $g(x)$.

Cookbook for horizontal shifts

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To translate the graph of this function horizontally rightward by $c$ units, replace "$x$" with "$x-c$". Optionally, simplify the resulting equation.
3. To translate a function horizontally leftward, use a negative number for $c$, then follow the procedure for a rightward shift. (For example, to move 2 units leftward, use $c=-2$.)
4. Finally, replace "$y$" with "$g(x)$".

Example: Move the graph of the square root function $f(x)=\sqrt{x}$ horizontally leftward by 4 units.

1. Substitute "$y$" for "$f(x)$": $$y=\sqrt{x}\mathrm{.}$$
   
2. Replace "$x$" with "$x-c$": $$y=\sqrt{x-c}\mathrm{.}$$
   
3. Since the shift is downward, set $c=-4$, then substitute and simplify: $$y=\sqrt{x-(-4)}\to y=\sqrt{x+4}\mathrm{.}$$
   
4. Replace "$y$" with "$g(x)$": $$g(x)=\sqrt{x+4}\mathrm{.}$$
   This is the equation of the shifted function. Since $f(x)=\sqrt{x}$, we may also write this in the form $$g(x)=f(x+4)\mathrm{,}$$
   which shows how a generic leftward shift may be written in terms of the original and shifted functions $f(x)$ and $g(x)$.

Additional examples of shifts for a generic function, including a combination of two shifts, are depicted in the following diagram.

Reflections

The effect of a reflection is to cause the graph of the reflected function to flip (rotate) either vertically or horizontally relative to the graph of the basic function to which it is related. The flip usually also results in a displacement of the graph across one axis, unless the original function was symmetric about the y-axis and the rotation also occurs about this axis. For example, the graph of the basic function $$f(x)=\sqrt{x}$$
is an increasing function whose graph (in Cartesian coordinates) lies entirely in the first quadrant, with the exception of a single point that coincides with the origin. To reflect this graph about the y-axis without deforming it so that the reflection lies in the second quadrant, we change the function to $$g(x)=\sqrt{-x}\mathrm{.}$$
To reflect this graph about the x-axis without deforming it so that the reflection lies in the fourth quadrant, we change the function to $$h(x)=-\sqrt{x}\mathrm{.}$$
Note that both types of reflections involve the insertion of a negative sign into the function. There are subtle distinctions between the locations of the insertions that distinguish between a reflection about the y-axis and a reflection about the x-axis. We discuss these distinctions in more detail below.

As in the instructions for translations, we assume that $f(x)$ represents the original function, and $g(x)$ represents the reflected function.

Cookbook for reflections about the y-axis

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To reflect the graph of this function about the y-axis, replace "$x$" with "$(-x)$" everywhere it occurs in the equation, then simplify the result, being sure to keep $y$ isolated.
3. Finally, replace "$y$" with "$g(x)$".

Example: Reflect the graph of the quadratic function $f(x)=x^2$ about the y-axis.

1. Substitute "$y$" for "$f(x)$": $$y=x^2\mathrm{.}$$
   
2. Replace "$x$" with "$(-x)$", then simplify: $$y={(-x)}^2\to y=x^2\mathrm{.}$$
   
3. Replace "$y$" with "$g(x)$": $$g(x)=x^2\mathrm{.}$$
   This is the equation of the reflected function. In this case, the original function was also symmetric about the y-axis, so the reflected function $g(x)$ is the same as the original function $f(x)$. The generic form of any reflection about the y-axis may be written $$g(x)=f(-x)\mathrm{.}$$

Cookbook for reflections about the x-axis

1. If the equation is written using function ($f(x)$) notation, replace "$f(x)$" with "$y$".
2. To reflect the graph of this function about the x-axis, replace "$y$" with "$(-y)$", then solve the resulting equation for $y$, usually by multiplying through both sides of the equation by −1.
3. Finally, replace "$y$" with "$g(x)$".

Example: Reflect the graph of the quadratic function $f(x)=x^2$ about the x-axis.

1. Substitute "$y$" for "$f(x)$": $$y=x^2\mathrm{.}$$
   
2. Replace "$y$" with "$(-y)$", then isolate $y$: $$(-y)=x^2\to y=-x^2\mathrm{.}$$
   
3. Replace "$y$" with "$g(x)$": $$g(x)=-x^2$$
   This is the equation of the reflected function. Since $$f(x)=x^2\mathrm{,}$$
   we may also write this in the form $$g(x)=-f(x)$$
   which shows how a generic reflection about the x-axis may be written in terms of the original and shifted functions $f(x)$ and $g(x)$.

Additional examples of reflections for a generic function, including a combination of two reflections, are depicted in the following diagram.

Deformations

The effect of a deformation is to cause the graph of the deformed function to expand (stretch) or contract (shrink) either vertically or horizontally relative to the graph of the basic function to which it is related. The deformation results in a change to the function's aspect ratio, causing the visual appearance of the deformed graph to become "stretched" or "squashed" in one direction or another. The effect is similar to what one might see when viewing one's reflection in a curved mirror, such as might be found in a carnival "fun house".

To deform a function's graph, we multiply either x or y by a positive number $c$ whose value is other than 1. (In principle, a negative number might also be used, but this would introduce a reflection in addition to the deformation; see above.) The type of deformation obtained (e.g., shrink or stretch) depends on the value of $c$ and on whether it multiplies x or y.

We have developed two diagrams to illustrate the effects of deformations. The first is devoted to vertical and horizontal stretches; the second to vertical and horizontal shrinks. Both diagrams employ the curve $$f(x)=x^3$$
as the basic function from which we derive all the deformations. To demonstrate how each deformation affects the domain or range of the derived functions, we limit the domain of $f(x)$ to the closed interval [−2, 2]. In addition, we specify the multiplicative constant $c$ to have a value of either 2 or $\frac{1}{2}\mathrm{,}$ which makes it easy to visualize the effects of each deformation on the functions' graphs; each stretch depicted expands the graph by a factor of 2, and each shrink depicted contracts the graph by the same factor (since multiplying by $\frac{1}{2}$ is equivalent to dividing by 2).

Stretches

The diagram appearing below will serve as the basis for the discussion of stretch deformations that follows.

Basic function

The basic function $$f(x)=x^3\mathrm{,}$$
limited to the domain [−2, 2], is depicted as the dark blue trace above.

Vertical stretch

The pink trace $$g(x)=2x^3$$
represents a vertical stretch deformation of this function by a factor of $$c=2.$$
For a vertical stretch, $c$ represents the factor by which the vertical aspect of the graph is expanded. Also note that for any vertical stretch, the domains of both the original and stretched functions are the same. However, the range of the stretched function is different from that of the original function, since the stretched function's graph extends to larger y values at its upper end, and smaller y values at its lower end.

The form of the equation for a generic vertical stretch of any function is $$g(x)=cf(x)$$
where $$c>1.$$

Horizontal stretch

The yellow trace $$h(x)=\frac{1}{8}{x}^3$$
represents a horizontal stretch deformation of this function, again by a factor of 2. However, in this case, $$c=\frac{1}{2}$$
For a horizontal stretch, $c$ represents the reciprocal of the factor by which the horizontal aspect of the graph is expanded. Also note that for any horizontal stretch, the ranges of both the original and stretched functions are the same. However, the domain of the stretched function is different from that of the original function, since the stretched function's graph extends to larger x values at its right end, and smaller x values at its left end.

The form of the equation for a generic horizontal stretch of any function is $$h(x)=f(cx)$$
where $$0<c<1.$$
In the specific case depicted here, we obtain the equation of the horizontally stretched function $h(x)$ given above from this generic form as follows: $$h(x)=f(cx)=f\left(\frac{1}{2}x\right)={\left(\frac{1}{2}x\right)}^3={\left(\frac{1}{2}\right)}^3{x}^3=\frac{1}{8}{x}^3$$

Combined stretch

The light blue trace $$k(x)=\frac{1}{4}{x}^3$$
represents the combination of horizontal and vertical stretches of this function, each by a factor of 2. The domain of this function matches the domain of the horizontally stretched function (yellow trace), and the range of this function matches the range of the vertically stretched function (pink trace).

Shrinks

The diagram appearing below will serve as the basis for the discussion of shrink deformations that follows.

Basic function

The basic function $$f(x)=x^3\mathrm{,}$$
limited to the domain [−2, 2], is depicted as the dark blue trace above. It is identical to the basic function employed in the description of vertical and horizontal stretches that appears above, and we have graphed it using the same scale as in the preceding diagram.

Vertical shrink

The pink trace $$g(x)=\frac{1}{2}{x}^3$$
represents a vertical shrink deformation of this function by a factor of $$c=\frac{1}{2}\mathrm{.}$$
As was true for a vertical stretch, $c$ represents the factor by which the vertical aspect of the graph is changed (contracted if $0<c<1$, or expanded if $c>1$). As was also true for a vertical stretch, the domains of both the original and deformed functions are the same, but the range of the deformed function is different from that of the original function, since the shrunken function's graph extends to smaller y values at its upper end, and larger y values at its lower end.

The form of the equation for a generic vertical shrink of any function is $$g(x)=cf(x)$$
where $$0<c<1.$$
This functional form is the same as for a vertical stretch; the only difference is that the multiplicative constant $c$ takes on a different (larger) set of values for a vertical stretch than for a vertical shrink.

Horizontal shrink

The yellow trace $$h(x)=8x^3$$
represents a horizontal shrink deformation of this function, again by a factor of 2. However, in this case, $$c=2$$
For a horizontal shrink, $c$ represents the reciprocal of the factor by which the horizontal aspect of the graph is contracted. As for the horizontal stretch, the ranges of both the original and shrunken functions are the same. However, the domain of the shrunken function is different from that of the original function, since the shrunken function's graph extends to smaller x values at its right end, and larger x values at its left end.

The form of the equation for a generic horizontal shrink of any function is $$h(x)=f(cx)$$
where $$c>1.$$
In the specific case depicted here, we obtain the equation of the horizontally stretched function $h(x)$ given above from this generic form as follows: $$h(x)=f(cx)=f\left(2x\right)={\left(2x\right)}^3={\left(2\right)}^3{x}^3=8x^3$$

Combined shrink

The light blue trace $$k(x)=4x^3$$
represents the combination of horizontal and vertical shrinks of this function, each by a factor of 2. The domain of this function matches the domain of the horizontally shrunken function (yellow trace), and the range of this function matches the range of the vertically shrunken function (pink trace).

Multiple transformations

More complicated transformations may be carried out by combining two or more of the simple transformations described above. The accompanying diagram illustrates a few of these possibilities. The legend documents the types of transformations that were combined to achieve each graph depicted on the coordinate grid. The generic functional forms are also provided in the legend.